

A306015


Exponential series expansion of (exp(x*y) + sinh(x)  cosh(x))/(1  x).


2



0, 1, 1, 1, 2, 1, 4, 6, 3, 1, 15, 24, 12, 4, 1, 76, 120, 60, 20, 5, 1, 455, 720, 360, 120, 30, 6, 1, 3186, 5040, 2520, 840, 210, 42, 7, 1, 25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1
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OFFSET

0,5


COMMENTS

For 0 <= k <= n, T(n,k) is the number of nonderangements of size n in which k of the fixed points are colored red. In particular, with D_n the derangement number A000166(n), T(n,0) = n!  D_n. For a general example, T(3,1) = 6 counts the colored permutations R23, R32, 1R3, 3R1, 12R, 21R where the red fixed points are indicated by "R".
For n >= k >= 1, T(n,k) = n!/k!. Proof. In a colored permutation, such as 3R7R516 counted by T(n,k) with n = 7 and k = 2, the R's indicate (red) fixed points and so no information is lost by rank ordering the remaining entries while retaining the placement of the R's: 2R5R314. The result is a permutation of the set consisting of 1,2,...,nk and k R's; there are n!/k! such permutations and the process is reversible. QED. (End)


LINKS



EXAMPLE

n  k = 0 1 2 3 4 5 6 7 8 9
+
0  0
1  1, 1
2  1, 2, 1
3  4, 6, 3, 1
4  15, 24, 12, 4, 1
5  76, 120, 60, 20, 5, 1
6  455, 720, 360, 120, 30, 6, 1
7  3186, 5040, 2520, 840, 210, 42, 7, 1
8  25487, 40320, 20160, 6720, 1680, 336, 56, 8, 1
9  229384, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1


MAPLE

gf := (exp(x*y) + sinh(x)  cosh(x))/(1  x):
ser := series(gf, x, 16): L := [seq(n!*coeff(ser, x, n), n=0..14)]:
seq(seq(coeff(L[k+1], y, n), n=0..k), k=0..12);


MATHEMATICA

Join[{0}, With[{nmax = 15}, CoefficientList[CoefficientList[Series[ (Exp[x*y] + Sinh[x]  Cosh[x])/(1  x), {x, 0, nmax}, {y, 0, nmax}], x], y ]*Range[0, nmax]!] // Flatten ] (* G. C. Greubel, Jul 18 2018 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



